Question

N = abcd is a four digit number and M = xyz is a three digit number. \(N \times M\) is:

• A. a five digit number when a = x = 1
• B. a six digit number when a = 3 and x = 3
• C. a seven digit number when a = 6 and x = 2
• D. an eight digit number when a = b = c = d = x = y = z = 9

Hint:

Fix the leading digits to get the smallest and largest possible product, then count digits at both ends of that range. Only option (3) keeps the same digit count throughout.

Answer 1

The Correct Option is C

Step 1: Set up the ranges.
N is a four digit number, so once we fix its leading digit a, N lies between \(a \times 1000\) and \(a \times 1000 + 999\). Similarly M is a three digit number, so once we fix its leading digit x, M lies between \(x \times 100\) and \(x \times 100 + 99\). We check each option by finding the smallest and largest possible product \(N \times M\) for the given leading digits, and counting how many digits that range spans.

Step 2: Check option (1), a = x = 1.
N ranges from 1000 to 1999, and M ranges from 100 to 199.
Smallest product: \(1000 \times 100 = 100000\), which has 6 digits.
Largest product: \(1999 \times 199 = 397801\), which also has 6 digits.
So the product is always a six digit number here, never five digits. Option (1) is wrong.

Step 3: Check option (2), a = 3 and x = 3.
N ranges from 3000 to 3999, and M ranges from 300 to 399.
Smallest product: \(3000 \times 300 = 900000\), 6 digits.
Largest product: \(3999 \times 399 = 1595601\), which is 7 digits.
Since the product can be either 6 or 7 digits depending on the exact values, it is not always a six digit number. Option (2) is wrong.

Step 4: Check option (3), a = 6 and x = 2.
N ranges from 6000 to 6999, and M ranges from 200 to 299.
Smallest product: \(6000 \times 200 = 1200000\), which has 7 digits.
Largest product: \(6999 \times 299 = 2092701\), which also has 7 digits.
Both ends of the range have exactly 7 digits, and since the product grows smoothly as N and M increase, every value in between also has 7 digits. So the product is always a seven digit number here. Option (3) is correct.

Step 5: Check option (4), all digits equal to 9.
Here \(N = 9999\) and \(M = 999\).
\[ 9999 \times 999 = 9999 \times 1000 - 9999 = 9999000 - 9999 = 9989001 \]
This has 7 digits, not 8. Option (4) is wrong.

Final Answer:
Only option (3) always gives a product with a fixed digit count across its whole range, a seven digit number. \[ \boxed{\text{Option (3): a seven digit number when } a=6, x=2} \]